Question

Let E and F be events for which P(E) = .5, P(F)= .4, and P(E $\cap$ F) = .2

a) are E and F mutually exclusive or independent? (justify mathematically)

b) Find P(E$\cup$ F)

c) Find P(F')

d) Find P(F l E)

e) Find P(E' $\cap$ F)

Answer 1

A) For a mutually exclusive event P(A and B) =0, but here P(E and F)= 0.2. So,can not be a exclusive events.

For a independent events P( A and B)= P(A) * P(B) and P(E and F) =P(E)*P(F)= 0.5*0.4=0.2. So, both are independent events.

B) P(A or B)= P(A) +P(B)- P(A and B)

P(E or F) = P(E) +P(F)- P(E and F) = 0.5+0.4-0.2= 0.7

C) P(F')

P(A') =1-P(A)

P(F')= 1-P(F) =1- 0.4= 0.6

D) P( F | E)

$P(F|E)= \frac{P(E \cap F)}{P(E)}=\frac{0.2}{0.5}=0.4=P(F)$

For an independent event: $P(F|E)= \frac{P(E \cap F)}{P(E)}=P(F)$

E) $P(E' \cap F)$

$P(A' \cap B)= P(B)-P(A\cap B)$

$P(E' \cap F)= P(F)-P(E\cap F)= 0.4-0.2=0.2$